Layer-Cake Formula for the Second Moment

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $Y$ be a random variable on it with $Y(\omega)\ge0$ for every $\omega\in\Omega$, and let $m$ denote Lebesgue measure on the Borel $\sigma$-algebra of $\mathbb{R}$. Then:

1. The pointwise square $Y^2$ is a nonnegative random variable, and the function $\varphi:\mathbb{R}\to\mathbb{R}$ defined by

is measurable with respect to the Borel $\sigma$-algebra.

2. With the Lebesgue integral of nonnegative measurable functions, whose value may be $+\infty$,

3. In particular, $Y$ is square-integrable if and only if $\int_{\mathbb{R}}\varphi\,dm<\infty$, and in that case the second moment satisfies